Lambda-Calculus and Combinators

An Introduction

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Hardcover | 24 July 2008 | Edition Number 2

At a Glance

Hardcover
358 Pages

Edition Type
Revised

Dimensions(cm)
23.7 x 15.9 x 2.4

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Combinatory logic and lambda-calculus, originally devised in the 1920's, have since developed into linguistic tools, especially useful in programming languages. The authors' previous book served as the main reference for introductory courses on lambda-calculus for over 20 years: this long-awaited new version is thoroughly revised and offers a fully up-to-date account of the subject, with the same authoritative exposition. The grammar and basic properties of both combinatory logic and lambda-calculus are discussed, followed by an introduction to type-theory. Typed and untyped versions of the systems, and their differences, are covered. Lambda-calculus models, which lie behind much of the semantics of programming languages, are also explained in depth. The treatment is as non-technical as possible, with the main ideas emphasized and illustrated by examples. Many exercises are included, from routine to advanced, with solutions to most at the end of the book.

Industry Reviews

From reviews of the 1st edition: 'The book of R. Hindley and J. Seldin is a very good introduction to fundamental techniques and results in these fields ... the book is clear, pleasant to read, and it needs no previous knowledge in the domain, but only basic notions of mathematical logic ... Clearly, it was impossible to treat everything in detail; but even when a subject is only skimmed, the book always provides an incentive for going deeper, and furnishes the means to do it, owing to a substantial bibliography. Several chapters end with interesting and useful notes with history, comments, and indications for further reading ... In conclusion, this book is very interesting and well written, and is highly recommended to everyone who wants to approach combinatory logic and lambda-calculus (logicians or computer scientists). J. Symbolic Logic 'The best general book on lambda-calculus (typed or untyped) and the theory of combinators.' Gerard Huet, INRIA

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Preface	p. ix
The [lambda]-calculus	p. 1
Introduction	p. 1
Term-structure and substitution	p. 5
[beta]-reduction	p. 11
[beta]-equality	p. 16
Combinatory logic	p. 21
Introduction to CL	p. 21
Weak reduction	p. 24
Abstraction in CL	p. 26
Weak equality	p. 29
The power of [lambda] and combinators	p. 33
Introduction	p. 33
The fixed-point theorem	p. 34
Bohm's theorem	p. 36
The quasi-leftmost-reduction theorem	p. 40
History and interpretation	p. 43
Representing the computable functions	p. 47
Introduction	p. 47
Primitive recursive functions	p. 50
Recursive functions	p. 56
Abstract numerals and Z	p. 61
The undecidability theorem	p. 63
The formal theories [lambda beta] and CLw	p. 69
The definitions of the theories	p. 69
First-order theories	p. 72
Equivalence of theories	p. 73
Extensionality in [lambda]-calculus	p. 76
Extensional equality	p. 76
[beta eta]-reduction in [lambda]-calculus	p. 79
Extensionality in combinatory logic	p. 82
Extensional equality	p. 82
Axioms for extensionality in CL	p. 85
Strong reduction	p. 89
Correspondence between [lambda] and CL	p. 92
Introduction	p. 92
The extensional equalities	p. 95
New abstraction algorithms in CL	p. 100
Combinatory [beta]-equality	p. 102
Simple typing, Church-style	p. 107
Simple types	p. 107
Typed [lambda]-calculus	p. 109
Typed CL	p. 115
Simple typing, Curry-style in CL	p. 119
Introduction	p. 119
The system [characters not reproducible]	p. 122
Subject-construction	p. 126
Abstraction	p. 127
Subject-reduction	p. 130
Typable CL-terms	p. 134
Link with Church's approach	p. 137
Principal types	p. 138
Adding new axioms	p. 143
Propositions-as-types and normalization	p. 147
The equality-rule Eq'	p. 155
Simple typing, Curry-style in [lambda]	p. 159
The system [characters not reproducible]	p. 159
Basic properties of [characters not reproducible]	p. 165
Typable [lambda]-terms	p. 170
Propositions-as-types and normalization	p. 173
The equality-rule Eq'	p. 176
Generalizations of typing	p. 180
Introduction	p. 180
Dependent function types, introduction	p. 181
Basic generalized typing, Curry-style in [lambda]	p. 183
Deductive rules to define types	p. 187
Church-style typing in [lambda]	p. 191
Normalization in PTSs	p. 202
Propositions-as-types	p. 209
PTSs with equality	p. 217
Models of CL	p. 220
Applicative structures	p. 220
Combinatory algebras	p. 223
Models of [lambda]-calculus	p. 229
The definition of [lambda]-model	p. 229
Syntax-free definitions	p. 236
General properties of [lambda]-models	p. 242
Scott's D[infinity] and other models	p. 247
Introduction: complete partial orders	p. 247
Continuous functions	p. 252
The construction of D[infinity]	p. 256
Basic properties of D[infinity]	p. 261
D[infinity] is a [lambda]-model	p. 267
Some other models	p. 271
Bound variables and [alpha]-conversion	p. 276
Confluence proofs	p. 282
Strong normalization proofs	p. 293
Care of your pet combinator	p. 305
Answers to starred exercises	p. 307
References	p. 323
List of symbols	p. 334
Index	p. 337
Table of Contents provided by Ingram. All Rights Reserved.

Lambda-Calculus and Combinators

An Introduction

At a Glance

Hardcover

Industry Reviews

Shipping

How to return your order

Defective items

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